LeetCode 834 Sum of Distances in Tree - Key Observation

When you move the root from node $p$ to its child $c$:

$1.$ Nodes in $c$'s subtree get $1$ closer (they're now below the root instead of $2$ edges away).

$2.$ All other nodes get $1$ farther (they have to go through $c$ now). If $c$'s subtree has $\text{count}[c]$ nodes, then $n - \text{count}[c]$ nodes get farther. So: $\text{answer}[c] = \text{answer}[p] - \text{count}[c] + (n - \text{count}[c]) = \text{answer}[p] + n - 2 \cdot \text{count}[c]$.